PDE in clairaut's form

Asked May 18 '17 at 15:20

Active May 18 '17 at 15:25

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A solution of the PDE

$$x \frac{\partial u}{\partial x} +y \frac{\partial u}{\partial y}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2-u=0 $$

represents

(a) an ellipse in the $XY$ plane

(b) an ellipsoid in the $XYU$ space

(c) a parabola in the $UX$ plane

(d) a hyperbola in the $UY$ plane

I know that it is in the $clairaut's$ form, so a solution should be of the form $u=ax+by+a^2+b^2$, $a, b$ are constants. But I am confused how this solution can satisfy any one of them.

edited Jun 12 '20 at 10:38

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asked May 18 '17 at 15:20

Hirakjyoti Das

That would be "none of the above". But there are also solutions of the form $u = a x^2 + b x y + c y^2$ for suitable $a,b,c$. – Robert Israel May 18 '17 at 15:38
Prof. Israel, for $ u= ax^2+ bxy+ cy^2$ could we tick option a and c with sutiable $a,b,c$ ? thanks prof. – Hirakjyoti Das May 18 '17 at 15:48
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The solution is not an ellipse in the xy plane, though its level curves can be. – Robert Israel May 18 '17 at 17:08

PDE in clairaut's form

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